Spaces of PL manifolds and categories of simple maps

An $h$-cobordism on a closed $m$-manifold $M$ (differentiable or PL) is a compact manifold $W$ of dimension $m+1$ whose boundary is identified with a topological union $M \sqcup M'$ of closed $m$-manifolds such that the inclusions $M \hookrightarrow W$ and $M' \hookrightarrow W$ are homotopy equivalences. The $h$-cobordism theorem of Smale says that $W$ is isomorphic to a product $M\times [0,1]$ if $M$ is simply connected and $m\geq 5$. Dropping the assumption of simply-connectivity, the classes of $h$-cobordisms on a closed connected $M$ for $m\geq 5$, up to diffeomorphism or PL-homeomorphism relative to $M$, are in bijection with elements of $Wh(\pi_1(M))$, the Whitehead group of the fundamental group of $M$. The Whitehead group is a direct summand of the algebraic K-group $K_1({\mathbb Z}[\pi_1(M)])$ of the group ring ${\mathbb Z}[\pi_1(M)]$, where the complementary summand is isomorphic to the abelianization of $\pi_1(M) \otimes {\mathbb Z}/2$, and $K_1(R)$ is the abelianization of the direct limit of the groups $GL_n(R)$, for a ring $R$. An $s$-cobordism is an $h$-cobordism which is isomorphic to a product $M\times [0,1]$, that is its Whitehead element is zero. This is equivalent to the inclusion $M\hookrightarrow W$ being a simple homotopy equivalence (the homotopy equivalence can be obtained by a sequence of collapsing/expansions of simplices of $W$ without touching the simplices of $M$).

This book is intended to give the proof of a stable parametrized $h$-cobordism theorem: for $M$ a compact manifold (either differentiable, PL or topological), the space of stable $h$-cobordisms on $M$ is naturally homotopy equivalent to the loop space of the Whitehead space of $M$. The classical theorems mentioned above correspond to the $\pi_0$ of these spaces. The stable $h$-cobordism space is defined in terms of manifold bundles, whereas the Whitehead space is defined in terms of the algebraic K-theory of the space $M$. This completes a program initiated by the first author, F. Waldhausen, about thirty years ago.

The book is largely self-contained and gives complete proofs. It starts with a large introduction which explains the connection between cobordism theory and algebraic K-theory. However all explanations and proofs are for readers with a working knowledge of algebraic topology. Therefore the target readers are researchers and PhD students in the area of algebraic topology.

Vicente Muñoz
Book details

The classical version of the $h$-cobordism theorem determines when an $h$-cobordism is isomorphic to a product in terms of some algebraic K-theory invariant called its Whitehead element. This book is intended to give the proof of a parametrized $h$-cobordism theorem, that is determining the homotopy type of the space of $h$-cobordisms on a closed manifold. This is given in terms of a space constructed by algebraic K-theory means. The classical case corresponds thus to the $\pi_0$ of this space.




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